*Definitions*: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.

Let the *fixed-point subspace* $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.

Let the *pointwise stabilizer subgroup* $G_{(X)}:=\{ g \in G \ \vert \ gx=x \ , \forall x \in X \}$.

Let $G$ be a finite group, $H$ a subgroup, $U$ and $V$ two irreducible complex representations of $G$.

Let the subgroup $L := G_{(U^H)} \cap G_{(V^H)}$.

*Question*: Is there an irreducible $W \le U \otimes V$ such that $G_{(W^H)} \cap G_{(V^H)}$ and $G_{(U^H)} \cap G_{(W^H)} \subset L$?